We consider two choices for the directions \(d\) . In Random Coordinate Proximal Gradient (RCPG) algorithm, the direction \(d\)  is computed by a proximal operator of \(\psi \)  restricted to the subspace \(U_{i}\) . In Random Coordinate Gradient Descent (RCGD) algorithm, \(d\)  is given by a multiple of a block of components of the gradient \(\nabla F(x_{k})\) . Let us also recall the definition of the Kurdyka-Lojasiewicz (KL) property [1]}.
